<?php

    /**
     * @package JAMA
     *    Class to obtain eigenvalues and eigenvectors of a real matrix.
     *    If A is symmetric, then A = V*D*V' where the eigenvalue matrix D
     *    is diagonal and the eigenvector matrix V is orthogonal (i.e.
     *    A = V.times(D.times(V.transpose())) and V.times(V.transpose())
     *    equals the identity matrix).
     *    If A is not symmetric, then the eigenvalue matrix D is block diagonal
     *    with the real eigenvalues in 1-by-1 blocks and any complex eigenvalues,
     *    lambda + i*mu, in 2-by-2 blocks, [lambda, mu; -mu, lambda].  The
     *    columns of V represent the eigenvectors in the sense that A*V = V*D,
     *    i.e. A.times(V) equals V.times(D).  The matrix V may be badly
     *    conditioned, or even singular, so the validity of the equation
     *    A = V*D*inverse(V) depends upon V.cond().
     * @author  Paul Meagher
     * @license PHP v3.0
     * @version 1.1
     */
    class EigenvalueDecomposition {
        /**
         *    Row and column dimension (square matrix).
         * @var int
         */
        private $n;
        /**
         *    Internal symmetry flag.
         * @var int
         */
        private $issymmetric;
        /**
         *    Arrays for internal storage of eigenvalues.
         * @var array
         */
        private $d = [];
        private $e = [];
        /**
         *    Array for internal storage of eigenvectors.
         * @var array
         */
        private $V = [];
        /**
         *    Array for internal storage of nonsymmetric Hessenberg form.
         * @var array
         */
        private $H = [];
        /**
         *    Working storage for nonsymmetric algorithm.
         * @var array
         */
        private $ort;
        /**
         *    Used for complex scalar division.
         * @var float
         */
        private $cdivr;
        private $cdivi;

        /**
         *    Constructor: Check for symmetry, then construct the eigenvalue decomposition
         * @access public
         * @param A  Square matrix
         * @return Structure to access D and V.
         */
        public function __construct($Arg) {
            $this->A = $Arg->getArray();
            $this->n = $Arg->getColumnDimension();
            $issymmetric = true;
            for ($j = 0; ($j < $this->n) & $issymmetric; ++$j) {
                for ($i = 0; ($i < $this->n) & $issymmetric; ++$i) {
                    $issymmetric = ($this->A[$i][$j] == $this->A[$j][$i]);
                }
            }
            if ($issymmetric) {
                $this->V = $this->A;
                // Tridiagonalize.
                $this->tred2();
                // Diagonalize.
                $this->tql2();
            } else {
                $this->H   = $this->A;
                $this->ort = [];
                // Reduce to Hessenberg form.
                $this->orthes();
                // Reduce Hessenberg to real Schur form.
                $this->hqr2();
            }
        }

        /**
         *    Symmetric Householder reduction to tridiagonal form.
         * @access private
         */
        private function tred2() {
            //  This is derived from the Algol procedures tred2 by
            //  Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
            //  Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
            //  Fortran subroutine in EISPACK.
            $this->d = $this->V[$this->n - 1];
            // Householder reduction to tridiagonal form.
            for ($i = $this->n - 1; $i > 0; --$i) {
                $i_ = $i - 1;
                // Scale to avoid under/overflow.
                $h     = $scale = 0.0;
                $scale += array_sum(array_map(abs, $this->d));
                if ($scale == 0.0) {
                    $this->e[$i] = $this->d[$i_];
                    $this->d     = array_slice($this->V[$i_], 0, $i_);
                    for ($j = 0; $j < $i; ++$j) {
                        $this->V[$j][$i] = $this->V[$i][$j] = 0.0;
                    }
                } else {
                    // Generate Householder vector.
                    for ($k = 0; $k < $i; ++$k) {
                        $this->d[$k] /= $scale;
                        $h           += pow($this->d[$k], 2);
                    }
                    $f = $this->d[$i_];
                    $g = sqrt($h);
                    if ($f > 0) {
                        $g = -$g;
                    }
                    $this->e[$i]  = $scale * $g;
                    $h            = $h - $f * $g;
                    $this->d[$i_] = $f - $g;
                    for ($j = 0; $j < $i; ++$j) {
                        $this->e[$j] = 0.0;
                    }
                    // Apply similarity transformation to remaining columns.
                    for ($j = 0; $j < $i; ++$j) {
                        $f               = $this->d[$j];
                        $this->V[$j][$i] = $f;
                        $g               = $this->e[$j] + $this->V[$j][$j] * $f;
                        for ($k = $j + 1; $k <= $i_; ++$k) {
                            $g           += $this->V[$k][$j] * $this->d[$k];
                            $this->e[$k] += $this->V[$k][$j] * $f;
                        }
                        $this->e[$j] = $g;
                    }
                    $f = 0.0;
                    for ($j = 0; $j < $i; ++$j) {
                        $this->e[$j] /= $h;
                        $f           += $this->e[$j] * $this->d[$j];
                    }
                    $hh = $f / (2 * $h);
                    for ($j = 0; $j < $i; ++$j) {
                        $this->e[$j] -= $hh * $this->d[$j];
                    }
                    for ($j = 0; $j < $i; ++$j) {
                        $f = $this->d[$j];
                        $g = $this->e[$j];
                        for ($k = $j; $k <= $i_; ++$k) {
                            $this->V[$k][$j] -= ($f * $this->e[$k] + $g * $this->d[$k]);
                        }
                        $this->d[$j]     = $this->V[$i - 1][$j];
                        $this->V[$i][$j] = 0.0;
                    }
                }
                $this->d[$i] = $h;
            }
            // Accumulate transformations.
            for ($i = 0; $i < $this->n - 1; ++$i) {
                $this->V[$this->n - 1][$i] = $this->V[$i][$i];
                $this->V[$i][$i]           = 1.0;
                $h                         = $this->d[$i + 1];
                if ($h != 0.0) {
                    for ($k = 0; $k <= $i; ++$k) {
                        $this->d[$k] = $this->V[$k][$i + 1] / $h;
                    }
                    for ($j = 0; $j <= $i; ++$j) {
                        $g = 0.0;
                        for ($k = 0; $k <= $i; ++$k) {
                            $g += $this->V[$k][$i + 1] * $this->V[$k][$j];
                        }
                        for ($k = 0; $k <= $i; ++$k) {
                            $this->V[$k][$j] -= $g * $this->d[$k];
                        }
                    }
                }
                for ($k = 0; $k <= $i; ++$k) {
                    $this->V[$k][$i + 1] = 0.0;
                }
            }
            $this->d                             = $this->V[$this->n - 1];
            $this->V[$this->n - 1]               = array_fill(0, $j, 0.0);
            $this->V[$this->n - 1][$this->n - 1] = 1.0;
            $this->e[0]                          = 0.0;
        }

        /**
         *    Symmetric tridiagonal QL algorithm.
         *    This is derived from the Algol procedures tql2, by
         *    Bowdler, Martin, Reinsch, and Wilkinson, Handbook for
         *    Auto. Comp., Vol.ii-Linear Algebra, and the corresponding
         *    Fortran subroutine in EISPACK.
         * @access private
         */
        private function tql2() {
            for ($i = 1; $i < $this->n; ++$i) {
                $this->e[$i - 1] = $this->e[$i];
            }
            $this->e[$this->n - 1] = 0.0;
            $f                     = 0.0;
            $tst1                  = 0.0;
            $eps                   = pow(2.0, -52.0);
            for ($l = 0; $l < $this->n; ++$l) {
                // Find small subdiagonal element
                $tst1 = max($tst1, abs($this->d[$l]) + abs($this->e[$l]));
                $m    = $l;
                while ($m < $this->n) {
                    if (abs($this->e[$m]) <= $eps * $tst1) {
                        break;
                    }
                    ++$m;
                }
                // If m == l, $this->d[l] is an eigenvalue,
                // otherwise, iterate.
                if ($m > $l) {
                    $iter = 0;
                    do {
                        // Could check iteration count here.
                        $iter += 1;
                        // Compute implicit shift
                        $g = $this->d[$l];
                        $p = ($this->d[$l + 1] - $g) / (2.0 * $this->e[$l]);
                        $r = hypo($p, 1.0);
                        if ($p < 0) {
                            $r *= -1;
                        }
                        $this->d[$l]     = $this->e[$l] / ($p + $r);
                        $this->d[$l + 1] = $this->e[$l] * ($p + $r);
                        $dl1             = $this->d[$l + 1];
                        $h               = $g - $this->d[$l];
                        for ($i = $l + 2; $i < $this->n; ++$i) {
                            $this->d[$i] -= $h;
                        }
                        $f += $h;
                        // Implicit QL transformation.
                        $p   = $this->d[$m];
                        $c   = 1.0;
                        $c2  = $c3 = $c;
                        $el1 = $this->e[$l + 1];
                        $s   = $s2 = 0.0;
                        for ($i = $m - 1; $i >= $l; --$i) {
                            $c3              = $c2;
                            $c2              = $c;
                            $s2              = $s;
                            $g               = $c * $this->e[$i];
                            $h               = $c * $p;
                            $r               = hypo($p, $this->e[$i]);
                            $this->e[$i + 1] = $s * $r;
                            $s               = $this->e[$i] / $r;
                            $c               = $p / $r;
                            $p               = $c * $this->d[$i] - $s * $g;
                            $this->d[$i + 1] = $h + $s * ($c * $g + $s * $this->d[$i]);
                            // Accumulate transformation.
                            for ($k = 0; $k < $this->n; ++$k) {
                                $h                   = $this->V[$k][$i + 1];
                                $this->V[$k][$i + 1] = $s * $this->V[$k][$i] + $c * $h;
                                $this->V[$k][$i]     = $c * $this->V[$k][$i] - $s * $h;
                            }
                        }
                        $p           = -$s * $s2 * $c3 * $el1 * $this->e[$l] / $dl1;
                        $this->e[$l] = $s * $p;
                        $this->d[$l] = $c * $p;
                        // Check for convergence.
                    } while (abs($this->e[$l]) > $eps * $tst1);
                }
                $this->d[$l] = $this->d[$l] + $f;
                $this->e[$l] = 0.0;
            }
            // Sort eigenvalues and corresponding vectors.
            for ($i = 0; $i < $this->n - 1; ++$i) {
                $k = $i;
                $p = $this->d[$i];
                for ($j = $i + 1; $j < $this->n; ++$j) {
                    if ($this->d[$j] < $p) {
                        $k = $j;
                        $p = $this->d[$j];
                    }
                }
                if ($k != $i) {
                    $this->d[$k] = $this->d[$i];
                    $this->d[$i] = $p;
                    for ($j = 0; $j < $this->n; ++$j) {
                        $p               = $this->V[$j][$i];
                        $this->V[$j][$i] = $this->V[$j][$k];
                        $this->V[$j][$k] = $p;
                    }
                }
            }
        }

        /**
         *    Nonsymmetric reduction to Hessenberg form.
         *    This is derived from the Algol procedures orthes and ortran,
         *    by Martin and Wilkinson, Handbook for Auto. Comp.,
         *    Vol.ii-Linear Algebra, and the corresponding
         *    Fortran subroutines in EISPACK.
         * @access private
         */
        private function orthes() {
            $low  = 0;
            $high = $this->n - 1;
            for ($m = $low + 1; $m <= $high - 1; ++$m) {
                // Scale column.
                $scale = 0.0;
                for ($i = $m; $i <= $high; ++$i) {
                    $scale = $scale + abs($this->H[$i][$m - 1]);
                }
                if ($scale != 0.0) {
                    // Compute Householder transformation.
                    $h = 0.0;
                    for ($i = $high; $i >= $m; --$i) {
                        $this->ort[$i] = $this->H[$i][$m - 1] / $scale;
                        $h             += $this->ort[$i] * $this->ort[$i];
                    }
                    $g = sqrt($h);
                    if ($this->ort[$m] > 0) {
                        $g *= -1;
                    }
                    $h             -= $this->ort[$m] * $g;
                    $this->ort[$m] -= $g;
                    // Apply Householder similarity transformation
                    // H = (I -u * u' / h) * H * (I -u * u') / h)
                    for ($j = $m; $j < $this->n; ++$j) {
                        $f = 0.0;
                        for ($i = $high; $i >= $m; --$i) {
                            $f += $this->ort[$i] * $this->H[$i][$j];
                        }
                        $f /= $h;
                        for ($i = $m; $i <= $high; ++$i) {
                            $this->H[$i][$j] -= $f * $this->ort[$i];
                        }
                    }
                    for ($i = 0; $i <= $high; ++$i) {
                        $f = 0.0;
                        for ($j = $high; $j >= $m; --$j) {
                            $f += $this->ort[$j] * $this->H[$i][$j];
                        }
                        $f = $f / $h;
                        for ($j = $m; $j <= $high; ++$j) {
                            $this->H[$i][$j] -= $f * $this->ort[$j];
                        }
                    }
                    $this->ort[$m]       = $scale * $this->ort[$m];
                    $this->H[$m][$m - 1] = $scale * $g;
                }
            }
            // Accumulate transformations (Algol's ortran).
            for ($i = 0; $i < $this->n; ++$i) {
                for ($j = 0; $j < $this->n; ++$j) {
                    $this->V[$i][$j] = ($i == $j ? 1.0 : 0.0);
                }
            }
            for ($m = $high - 1; $m >= $low + 1; --$m) {
                if ($this->H[$m][$m - 1] != 0.0) {
                    for ($i = $m + 1; $i <= $high; ++$i) {
                        $this->ort[$i] = $this->H[$i][$m - 1];
                    }
                    for ($j = $m; $j <= $high; ++$j) {
                        $g = 0.0;
                        for ($i = $m; $i <= $high; ++$i) {
                            $g += $this->ort[$i] * $this->V[$i][$j];
                        }
                        // Double division avoids possible underflow
                        $g = ($g / $this->ort[$m]) / $this->H[$m][$m - 1];
                        for ($i = $m; $i <= $high; ++$i) {
                            $this->V[$i][$j] += $g * $this->ort[$i];
                        }
                    }
                }
            }
        }

        /**
         *    Nonsymmetric reduction from Hessenberg to real Schur form.
         *    Code is derived from the Algol procedure hqr2,
         *    by Martin and Wilkinson, Handbook for Auto. Comp.,
         *    Vol.ii-Linear Algebra, and the corresponding
         *    Fortran subroutine in EISPACK.
         * @access private
         */
        private function hqr2() {
            //  Initialize
            $nn      = $this->n;
            $n       = $nn - 1;
            $low     = 0;
            $high    = $nn - 1;
            $eps     = pow(2.0, -52.0);
            $exshift = 0.0;
            $p       = $q = $r = $s = $z = 0;
            // Store roots isolated by balanc and compute matrix norm
            $norm = 0.0;
            for ($i = 0; $i < $nn; ++$i) {
                if (($i < $low) or ($i > $high)) {
                    $this->d[$i] = $this->H[$i][$i];
                    $this->e[$i] = 0.0;
                }
                for ($j = max($i - 1, 0); $j < $nn; ++$j) {
                    $norm = $norm + abs($this->H[$i][$j]);
                }
            }
            // Outer loop over eigenvalue index
            $iter = 0;
            while ($n >= $low) {
                // Look for single small sub-diagonal element
                $l = $n;
                while ($l > $low) {
                    $s = abs($this->H[$l - 1][$l - 1]) + abs($this->H[$l][$l]);
                    if ($s == 0.0) {
                        $s = $norm;
                    }
                    if (abs($this->H[$l][$l - 1]) < $eps * $s) {
                        break;
                    }
                    --$l;
                }
                // Check for convergence
                // One root found
                if ($l == $n) {
                    $this->H[$n][$n] = $this->H[$n][$n] + $exshift;
                    $this->d[$n]     = $this->H[$n][$n];
                    $this->e[$n]     = 0.0;
                    --$n;
                    $iter = 0;
                    // Two roots found
                } elseif ($l == $n - 1) {
                    $w                       = $this->H[$n][$n - 1] * $this->H[$n - 1][$n];
                    $p                       = ($this->H[$n - 1][$n - 1] - $this->H[$n][$n]) / 2.0;
                    $q                       = $p * $p + $w;
                    $z                       = sqrt(abs($q));
                    $this->H[$n][$n]         = $this->H[$n][$n] + $exshift;
                    $this->H[$n - 1][$n - 1] = $this->H[$n - 1][$n - 1] + $exshift;
                    $x                       = $this->H[$n][$n];
                    // Real pair
                    if ($q >= 0) {
                        if ($p >= 0) {
                            $z = $p + $z;
                        } else {
                            $z = $p - $z;
                        }
                        $this->d[$n - 1] = $x + $z;
                        $this->d[$n]     = $this->d[$n - 1];
                        if ($z != 0.0) {
                            $this->d[$n] = $x - $w / $z;
                        }
                        $this->e[$n - 1] = 0.0;
                        $this->e[$n]     = 0.0;
                        $x               = $this->H[$n][$n - 1];
                        $s               = abs($x) + abs($z);
                        $p               = $x / $s;
                        $q               = $z / $s;
                        $r               = sqrt($p * $p + $q * $q);
                        $p               = $p / $r;
                        $q               = $q / $r;
                        // Row modification
                        for ($j = $n - 1; $j < $nn; ++$j) {
                            $z                   = $this->H[$n - 1][$j];
                            $this->H[$n - 1][$j] = $q * $z + $p * $this->H[$n][$j];
                            $this->H[$n][$j]     = $q * $this->H[$n][$j] - $p * $z;
                        }
                        // Column modification
                        for ($i = 0; $i <= $n; ++$i) {
                            $z                   = $this->H[$i][$n - 1];
                            $this->H[$i][$n - 1] = $q * $z + $p * $this->H[$i][$n];
                            $this->H[$i][$n]     = $q * $this->H[$i][$n] - $p * $z;
                        }
                        // Accumulate transformations
                        for ($i = $low; $i <= $high; ++$i) {
                            $z                   = $this->V[$i][$n - 1];
                            $this->V[$i][$n - 1] = $q * $z + $p * $this->V[$i][$n];
                            $this->V[$i][$n]     = $q * $this->V[$i][$n] - $p * $z;
                        }
                        // Complex pair
                    } else {
                        $this->d[$n - 1] = $x + $p;
                        $this->d[$n]     = $x + $p;
                        $this->e[$n - 1] = $z;
                        $this->e[$n]     = -$z;
                    }
                    $n    = $n - 2;
                    $iter = 0;
                    // No convergence yet
                } else {
                    // Form shift
                    $x = $this->H[$n][$n];
                    $y = 0.0;
                    $w = 0.0;
                    if ($l < $n) {
                        $y = $this->H[$n - 1][$n - 1];
                        $w = $this->H[$n][$n - 1] * $this->H[$n - 1][$n];
                    }
                    // Wilkinson's original ad hoc shift
                    if ($iter == 10) {
                        $exshift += $x;
                        for ($i = $low; $i <= $n; ++$i) {
                            $this->H[$i][$i] -= $x;
                        }
                        $s = abs($this->H[$n][$n - 1]) + abs($this->H[$n - 1][$n - 2]);
                        $x = $y = 0.75 * $s;
                        $w = -0.4375 * $s * $s;
                    }
                    // MATLAB's new ad hoc shift
                    if ($iter == 30) {
                        $s = ($y - $x) / 2.0;
                        $s = $s * $s + $w;
                        if ($s > 0) {
                            $s = sqrt($s);
                            if ($y < $x) {
                                $s = -$s;
                            }
                            $s = $x - $w / (($y - $x) / 2.0 + $s);
                            for ($i = $low; $i <= $n; ++$i) {
                                $this->H[$i][$i] -= $s;
                            }
                            $exshift += $s;
                            $x       = $y = $w = 0.964;
                        }
                    }
                    // Could check iteration count here.
                    $iter = $iter + 1;
                    // Look for two consecutive small sub-diagonal elements
                    $m = $n - 2;
                    while ($m >= $l) {
                        $z = $this->H[$m][$m];
                        $r = $x - $z;
                        $s = $y - $z;
                        $p = ($r * $s - $w) / $this->H[$m + 1][$m] + $this->H[$m][$m + 1];
                        $q = $this->H[$m + 1][$m + 1] - $z - $r - $s;
                        $r = $this->H[$m + 2][$m + 1];
                        $s = abs($p) + abs($q) + abs($r);
                        $p = $p / $s;
                        $q = $q / $s;
                        $r = $r / $s;
                        if ($m == $l) {
                            break;
                        }
                        if (abs($this->H[$m][$m - 1]) * (abs($q) + abs($r)) < $eps * (abs($p) * (abs($this->H[$m - 1][$m - 1]) + abs($z) + abs($this->H[$m + 1][$m + 1])))) {
                            break;
                        }
                        --$m;
                    }
                    for ($i = $m + 2; $i <= $n; ++$i) {
                        $this->H[$i][$i - 2] = 0.0;
                        if ($i > $m + 2) {
                            $this->H[$i][$i - 3] = 0.0;
                        }
                    }
                    // Double QR step involving rows l:n and columns m:n
                    for ($k = $m; $k <= $n - 1; ++$k) {
                        $notlast = ($k != $n - 1);
                        if ($k != $m) {
                            $p = $this->H[$k][$k - 1];
                            $q = $this->H[$k + 1][$k - 1];
                            $r = ($notlast ? $this->H[$k + 2][$k - 1] : 0.0);
                            $x = abs($p) + abs($q) + abs($r);
                            if ($x != 0.0) {
                                $p = $p / $x;
                                $q = $q / $x;
                                $r = $r / $x;
                            }
                        }
                        if ($x == 0.0) {
                            break;
                        }
                        $s = sqrt($p * $p + $q * $q + $r * $r);
                        if ($p < 0) {
                            $s = -$s;
                        }
                        if ($s != 0) {
                            if ($k != $m) {
                                $this->H[$k][$k - 1] = -$s * $x;
                            } elseif ($l != $m) {
                                $this->H[$k][$k - 1] = -$this->H[$k][$k - 1];
                            }
                            $p = $p + $s;
                            $x = $p / $s;
                            $y = $q / $s;
                            $z = $r / $s;
                            $q = $q / $p;
                            $r = $r / $p;
                            // Row modification
                            for ($j = $k; $j < $nn; ++$j) {
                                $p = $this->H[$k][$j] + $q * $this->H[$k + 1][$j];
                                if ($notlast) {
                                    $p                   = $p + $r * $this->H[$k + 2][$j];
                                    $this->H[$k + 2][$j] = $this->H[$k + 2][$j] - $p * $z;
                                }
                                $this->H[$k][$j]     = $this->H[$k][$j] - $p * $x;
                                $this->H[$k + 1][$j] = $this->H[$k + 1][$j] - $p * $y;
                            }
                            // Column modification
                            for ($i = 0; $i <= min($n, $k + 3); ++$i) {
                                $p = $x * $this->H[$i][$k] + $y * $this->H[$i][$k + 1];
                                if ($notlast) {
                                    $p                   = $p + $z * $this->H[$i][$k + 2];
                                    $this->H[$i][$k + 2] = $this->H[$i][$k + 2] - $p * $r;
                                }
                                $this->H[$i][$k]     = $this->H[$i][$k] - $p;
                                $this->H[$i][$k + 1] = $this->H[$i][$k + 1] - $p * $q;
                            }
                            // Accumulate transformations
                            for ($i = $low; $i <= $high; ++$i) {
                                $p = $x * $this->V[$i][$k] + $y * $this->V[$i][$k + 1];
                                if ($notlast) {
                                    $p                   = $p + $z * $this->V[$i][$k + 2];
                                    $this->V[$i][$k + 2] = $this->V[$i][$k + 2] - $p * $r;
                                }
                                $this->V[$i][$k]     = $this->V[$i][$k] - $p;
                                $this->V[$i][$k + 1] = $this->V[$i][$k + 1] - $p * $q;
                            }
                        }  // ($s != 0)
                    }  // k loop
                }  // check convergence
            }  // while ($n >= $low)
            // Backsubstitute to find vectors of upper triangular form
            if ($norm == 0.0) {
                return;
            }
            for ($n = $nn - 1; $n >= 0; --$n) {
                $p = $this->d[$n];
                $q = $this->e[$n];
                // Real vector
                if ($q == 0) {
                    $l               = $n;
                    $this->H[$n][$n] = 1.0;
                    for ($i = $n - 1; $i >= 0; --$i) {
                        $w = $this->H[$i][$i] - $p;
                        $r = 0.0;
                        for ($j = $l; $j <= $n; ++$j) {
                            $r = $r + $this->H[$i][$j] * $this->H[$j][$n];
                        }
                        if ($this->e[$i] < 0.0) {
                            $z = $w;
                            $s = $r;
                        } else {
                            $l = $i;
                            if ($this->e[$i] == 0.0) {
                                if ($w != 0.0) {
                                    $this->H[$i][$n] = -$r / $w;
                                } else {
                                    $this->H[$i][$n] = -$r / ($eps * $norm);
                                }
                                // Solve real equations
                            } else {
                                $x               = $this->H[$i][$i + 1];
                                $y               = $this->H[$i + 1][$i];
                                $q               = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i];
                                $t               = ($x * $s - $z * $r) / $q;
                                $this->H[$i][$n] = $t;
                                if (abs($x) > abs($z)) {
                                    $this->H[$i + 1][$n] = (-$r - $w * $t) / $x;
                                } else {
                                    $this->H[$i + 1][$n] = (-$s - $y * $t) / $z;
                                }
                            }
                            // Overflow control
                            $t = abs($this->H[$i][$n]);
                            if (($eps * $t) * $t > 1) {
                                for ($j = $i; $j <= $n; ++$j) {
                                    $this->H[$j][$n] = $this->H[$j][$n] / $t;
                                }
                            }
                        }
                    }
                    // Complex vector
                } elseif ($q < 0) {
                    $l = $n - 1;
                    // Last vector component imaginary so matrix is triangular
                    if (abs($this->H[$n][$n - 1]) > abs($this->H[$n - 1][$n])) {
                        $this->H[$n - 1][$n - 1] = $q / $this->H[$n][$n - 1];
                        $this->H[$n - 1][$n]     = -($this->H[$n][$n] - $p) / $this->H[$n][$n - 1];
                    } else {
                        $this->cdiv(0.0, -$this->H[$n - 1][$n], $this->H[$n - 1][$n - 1] - $p, $q);
                        $this->H[$n - 1][$n - 1] = $this->cdivr;
                        $this->H[$n - 1][$n]     = $this->cdivi;
                    }
                    $this->H[$n][$n - 1] = 0.0;
                    $this->H[$n][$n]     = 1.0;
                    for ($i = $n - 2; $i >= 0; --$i) {
                        // double ra,sa,vr,vi;
                        $ra = 0.0;
                        $sa = 0.0;
                        for ($j = $l; $j <= $n; ++$j) {
                            $ra = $ra + $this->H[$i][$j] * $this->H[$j][$n - 1];
                            $sa = $sa + $this->H[$i][$j] * $this->H[$j][$n];
                        }
                        $w = $this->H[$i][$i] - $p;
                        if ($this->e[$i] < 0.0) {
                            $z = $w;
                            $r = $ra;
                            $s = $sa;
                        } else {
                            $l = $i;
                            if ($this->e[$i] == 0) {
                                $this->cdiv(-$ra, -$sa, $w, $q);
                                $this->H[$i][$n - 1] = $this->cdivr;
                                $this->H[$i][$n]     = $this->cdivi;
                            } else {
                                // Solve complex equations
                                $x  = $this->H[$i][$i + 1];
                                $y  = $this->H[$i + 1][$i];
                                $vr = ($this->d[$i] - $p) * ($this->d[$i] - $p) + $this->e[$i] * $this->e[$i] - $q * $q;
                                $vi = ($this->d[$i] - $p) * 2.0 * $q;
                                if ($vr == 0.0 & $vi == 0.0) {
                                    $vr = $eps * $norm * (abs($w) + abs($q) + abs($x) + abs($y) + abs($z));
                                }
                                $this->cdiv($x * $r - $z * $ra + $q * $sa, $x * $s - $z * $sa - $q * $ra, $vr, $vi);
                                $this->H[$i][$n - 1] = $this->cdivr;
                                $this->H[$i][$n]     = $this->cdivi;
                                if (abs($x) > (abs($z) + abs($q))) {
                                    $this->H[$i + 1][$n - 1] = (-$ra - $w * $this->H[$i][$n - 1] + $q * $this->H[$i][$n]) / $x;
                                    $this->H[$i + 1][$n]     = (-$sa - $w * $this->H[$i][$n] - $q * $this->H[$i][$n - 1]) / $x;
                                } else {
                                    $this->cdiv(-$r - $y * $this->H[$i][$n - 1], -$s - $y * $this->H[$i][$n], $z, $q);
                                    $this->H[$i + 1][$n - 1] = $this->cdivr;
                                    $this->H[$i + 1][$n]     = $this->cdivi;
                                }
                            }
                            // Overflow control
                            $t = max(abs($this->H[$i][$n - 1]), abs($this->H[$i][$n]));
                            if (($eps * $t) * $t > 1) {
                                for ($j = $i; $j <= $n; ++$j) {
                                    $this->H[$j][$n - 1] = $this->H[$j][$n - 1] / $t;
                                    $this->H[$j][$n]     = $this->H[$j][$n] / $t;
                                }
                            }
                        } // end else
                    } // end for
                } // end else for complex case
            } // end for
            // Vectors of isolated roots
            for ($i = 0; $i < $nn; ++$i) {
                if ($i < $low | $i > $high) {
                    for ($j = $i; $j < $nn; ++$j) {
                        $this->V[$i][$j] = $this->H[$i][$j];
                    }
                }
            }
            // Back transformation to get eigenvectors of original matrix
            for ($j = $nn - 1; $j >= $low; --$j) {
                for ($i = $low; $i <= $high; ++$i) {
                    $z = 0.0;
                    for ($k = $low; $k <= min($j, $high); ++$k) {
                        $z = $z + $this->V[$i][$k] * $this->H[$k][$j];
                    }
                    $this->V[$i][$j] = $z;
                }
            }
        } // end hqr2

        /**
         *    Performs complex division.
         * @access private
         */
        private function cdiv($xr, $xi, $yr, $yi) {
            if (abs($yr) > abs($yi)) {
                $r           = $yi / $yr;
                $d           = $yr + $r * $yi;
                $this->cdivr = ($xr + $r * $xi) / $d;
                $this->cdivi = ($xi - $r * $xr) / $d;
            } else {
                $r           = $yr / $yi;
                $d           = $yi + $r * $yr;
                $this->cdivr = ($r * $xr + $xi) / $d;
                $this->cdivi = ($r * $xi - $xr) / $d;
            }
        }

        /**
         *    Return the eigenvector matrix
         * @access public
         * @return V
         */
        public function getV() {
            return new Matrix($this->V, $this->n, $this->n);
        }

        /**
         *    Return the real parts of the eigenvalues
         * @access public
         * @return real(diag(D))
         */
        public function getRealEigenvalues() {
            return $this->d;
        }

        /**
         *    Return the imaginary parts of the eigenvalues
         * @access public
         * @return imag(diag(D))
         */
        public function getImagEigenvalues() {
            return $this->e;
        }

        /**
         *    Return the block diagonal eigenvalue matrix
         * @access public
         * @return D
         */
        public function getD() {
            for ($i = 0; $i < $this->n; ++$i) {
                $D[$i]     = array_fill(0, $this->n, 0.0);
                $D[$i][$i] = $this->d[$i];
                if ($this->e[$i] == 0) {
                    continue;
                }
                $o         = ($this->e[$i] > 0) ? $i + 1 : $i - 1;
                $D[$i][$o] = $this->e[$i];
            }
            return new Matrix($D);
        }
    }
